Modal logic is a kind of logic used to represent statements about necessity and possibility. It plays a major role in philosophy and related fields as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula can be used to represent the statement that is known. In deontic modal logic, that same formula can represent that is a moral obligation.
Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false.
Modal logics are formal systems that include unary operators such as and , representing possibility and necessity respectively. For instance the modal formula can be read as "possibly " while can be read as "necessarily ". In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular, is true at a world if is true at some accessible possible world, while is true at a world if is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial.
While the intuition behind modal logic dates back to antiquity, the first modal axiomatic systems were developed by C. I. Lewis in 1912. The now-standard relational semantics emerged in the mid twentieth century from work by Arthur Prior, Jaakko Hintikka, and Saul Kripke. Recent developments include alternative topological semantics such as neighborhood semantics as well as applications of the relational semantics beyond its original philosophical motivation.[1] Such applications include game theory,[2] moral and legal theory,[2] web design,[2] multiverse-based set theory,[3] and social epistemology.[4]
^Blackburn, Patrick; de Rijke, Maarten; Venema, Yde (2001). Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press. ISBN 9780521527149.
^ abcvan Benthem, Johan (2010). Modal Logic for Open Minds(PDF). CSLI. S2CID 62162288. Archived from the original (PDF) on 2020-02-19.
^Hamkins, Joel (2012). "The set-theoretic multiverse". The Review of Symbolic Logic. 5 (3): 416–449. arXiv:1108.4223. doi:10.1017/S1755020311000359. S2CID 33807508.
^Baltag, Alexandru; Christoff, Zoe; Rendsvig, Rasmus; Smets, Sonja (2019). "Dynamic Epistemic Logics of Diffusion and Prediction in Social Networks". Studia Logica. 107 (3): 489–531. doi:10.1007/s11225-018-9804-x. S2CID 13968166.
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